Vibronic Coupling in Radicals: Is Quantum Chemistry Worth a Damn?

Vibronic Coupling in Radicals: Is Quantum Chemistry worth a Damn?

John F. Stanton University of Texas at Austin

Ohio State Spectroscopy Symposium, 2009 A typical problem in this area:

Experimentally determined levels of the asymmetric stretch

Large positive anharmonicity !!!!!!!!

Not consistent with otherwise reasonable-looking DFT curve. 2 Σg

5823 cm-1 2 Σu B N B anion - + -

B N B B N B - + - - + -

HOMO-1 (σg) HOMO (σu) 2 Σu adiabatic surface 2 Σg adiabatic surface How should a theoretical chemist think about this subject? How should a theoretical chemist think about this subject?

The molecular wavefunction:

Born-Huang expansion How should a theoretical chemist think about this subject?

The molecular wavefunction:

“quantum chemistry”

Born-Huang expansion How should a theoretical chemist think about this subject?

The molecular wavefunction:

“quantum chemistry”

Born-Huang expansion The usual choice is the adiabatic basis

Electronic Hamiltonian is diagonal in this basis for all possible nuclear coordinates R.

A pedagogically (but not computationally) useful choice is the static Born-Huang basis

Nuclear kinetic energy operator is diagonal in this basis. (Rigorous diabatic basis).

How best to think about these choices? SBH Adiabatic

FCI

Adiabatic Potential (PES)

TN V The usual assumption

TN V

(Tii + VPES,i) Ω = Evib Ω But sometimes… SBH Adiabatic

FCI

TN V But sometimes… SBH Adiabatic

FCI

TN V Alternative approach

Block diagonalize

Eigenvectors define TN V “quasidiabatic” electronic wavefunctions KDC “quasidiabatic” Hamiltonian

H. Köppel, W. Domcke and L.S. Cederbaum Adv. Chem. Phys. 57, 59 (1984) W. Domcke, H. Köppel and L.S. Cederbaum Mol. Phys. 43, 851 (1981)

See also: R.L. Fulton and M. Gouterman J. Chem. Phys. 35, 1059 (1961).

Recent developments: M. Nooijen (Waterloo), D. Yarkony (Johns Hopkins), UT Pros and Cons of the LVC model in this parametrization:

Extremely simple and outstanding qualitative model that has been used for nearly thirty years to identify vibronic coupling in molecular systems. Calculations are straightforward.

But…

Does not account for either anharmonicity or Duschinsky mixing in the symmetric modes.

Two state model approach for coupling coordinates relies on assumption that “quasidiabatic force constants” for coupling modes are similar in the two states. This is chemically reasonable (unlike similar assumptions about adiabatic force constants), but at best qualitatively accurate. And there are cases where this

assumption clearly fails (wagging mode of HCO2).

There are potential problems when the number of states involved in the coupling is greater than two (cyclopentadienyl radical) How can we make this quantitative? How “good” can it be?

Totally symmetric modes can be treated by a higher-order Taylor series. quadratic has been done sometimes (the so-called QVC) model, but we have gone up to quartic terms.

Typical parametrizations are such that the adiabatic and ab initio potentials are identical at the origin of the coordinate system (vertical parametrization). We have developed* a parametrization method such that the adiabatic model potentials agree exactly with the ab initio potentials at the minima of the final states (adiabatic parametrization).

By making an ansatz for the quasidiabatic states, we have recently come up with an analytic method for calculating the coupling constants**. This allows us to extend the coupling to nonlinear terms.

* T. Ichino, A.J. Gianola, W.C. Lineberger and J.F. Stanton, JCP 125, 084312 (2006). **T. Ichino, J. Gauss and J.F. Stanton, JCP, in press. Diabatic Diabatic Model H Model H (LVC) (elaborate) Expt.* 797 855 1961 2052 3153 3291 4443 5186 5888

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*K.R. Asmis, T.R. Taylor and D.M. Neumark JCP 111, 8838 (1999) Diabatic Diabatic Model H Model H (LVC) (elaborate) Expt.* 797 849 855 1961 2039 2052 3153 3278 3291 4443 4545 5186 5866 5888

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*K.R. Asmis, T.R. Taylor and D.M. Neumark JCP 111, 8838 (1999) * * Table II. DCO2¯ SEVI spectra peak positions, shifts from origin and PAD as well as the calculated frequencies and assignments. Peak eBE Shift from PADa Calculated Assignmentb origin position (cm-1) (cm-1) (cm-1) 0 2 A 28362 0 p 0.0 00 ( A1) 0 2 B 28449 87 s 100.5 00 ( B2) 1 2 C 28792 430 s 399.8 50 ( A1) 1 2 D 28925 563 p 564.8 30 ( A1) 1 2 E 29028 666 s 677.8 30 ( B2) 2 1 2 910.0 3050 ( A1) F 29310 948 s 970.2 1 2 50 ( B2) 2 2 G 29446 1084 p 1068.3 30 ( A1) 1 2 H 29498 1136 p 1138.6 20 ( A1) 2 2 I 29555 1193 s 1199.7 30 ( B2) 1 2 1478.1 10 ( A1) J 29882 1520 p 1492.6 1 2 2 3050 ( A1) K 30105 1743 s 1770.0 highly mixed L 30153 1791 s 1835.9 highly mixed M 30385 2023 p 1997.4 highly mixed N 30780 2418 O 30945 2583 p

E. Garand et al. JPC, in preparation. Calculated fundamental for “asymmetric stretch” was ca. 930 cm-1, about 50 cm-1 above experiment. Calculated fundamental for “asymmetric stretch” was ca. 930 cm-1, about 50 cm-1 above experiment. Just how accurate are adiabatic vibrational levels?

In other words:

“If I were to do a full CI calculation in a complete basis, and solve the vibrational problem exactly, would the results agree with experiment?”

Partial answer (and unsatisfying) answer:

“For water, it has been amply demonstrated that one can get -1 accuracies of ca. 1 cm by such an approach. But for NO3 and other strongly coupled systems, who knows?” Agree with me that HKDC “works” Agree with me that HKDC “works”

Simply ignore this “Exact” Adiabatic Experiment 849 855 2039 2052 3278 3291 4545 5866 5888

2492 2479 “Exact” Adiabatic Experiment 849 919 855 2039 2108 2052 3278 3367 3291 4545 5866 5888

2492 2479 “Exact” Adiabatic Experiment 849 919 855 2039 2108 2052 3272 3367 3291 4545 5846 5888

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WOW All done…

Thanks to: NSF, DOE and the Welch Foundation Many members of the audience Polynomial expansion

Very anharmonic!

Levels from one-dimensional Sch. Eqn. (DVR) Level Diff. Expt*

21 781.2 781.2 855

22 1914.3 1133.1 2052

23 3106.6 1192.3 3291

24 4378.4 1271.8 *K.R. Asmis, T.R. Taylor and D.M. Neumark JCP 111, 8838 (1999) Postdocs s Michael Harding Takatoshi Ichino Juana Vazquez

Graduate Students Undergrads Harshal Gupta Eric Buchanan David Price Devin Matthews Chris Simmons Ben McKown Mychel Varner Postdocs s Michael Harding Takatoshi Ichino Juana Vazquez

Graduate Students Undergrads Harshal Gupta Eric Buchanan David Price Devin Matthews Chris Simmons Ben McKown Mychel Varner

DOE-BES, NSF, Welch Foundation Towards more quantitative accuracy…

A more elaborate Hamiltonian Towards more quantitative accuracy…

A more elaborate Hamiltonian Outline

I. Vibronic Coupling in the Context of Negative Ion Photoelectron Spectroscopy.

II. Problems Encountered in Theoretical Descriptions Of Vibronic Effects: “Real v. Artifactual”.

III. Diabatic v. Adiabatic Pictures: The BNB Model System.

IV. Accurate Energy Level Positions for a Strongly Coupled System: Which Approach is Intrinsically More Accurate?

V. Application to Spectra of HCO2 and DCO2. Selection rules in photodetachment spectroscopy

Born-Oppenheimer

Crude (Koopmans’) MO model

Photodetachment Franck-Condon Cross-section Factor (FCF) General guidelines based on approximate models

1. Electrons can be detached from any electronic state (provided sufficiently energetic photons are supplied), and angular distribution of outgoing electron can be inferred.

2. If electron is detached from the ground state of the anion, only totally symmetric vibrations of the neutral should be seen in spectrum. Those which are most prominent are those associated with large changes in geometry between the anion and the neutral.

A tool that gives structural, symmetry and thermodynamic information about reactive molecular species. K.M. Ervin and W.C. Lineberger, JPC 95, 1167 (1991). Causes of breakdown of Franck-Condon approximation

1. Mixing of vibrational and electronic wavefunctions

(VIBRONIC COUPLING)

2. In this case, the molecular wavefunction of the bending level is “mixed” with that of the excited electronic state.

Vibronic coupling constant

A tool that gives structural, symmetry and thermodynamic information about reactive molecular species. What causes “artifactual” symmetry breaking?

The Hartree-Fock Hamiltonian, unlike the exact Born-Oppenheimer Electronic Hamiltonian does not commute with the symmetry operations associated with the point group of the nuclear framework

The Hartree-Fock wavefunction, again unlike the exact electronic (Born-Oppenheimer clamped-nucleus) wavefunction, does not have to transform as a pure irreducible representation of the point group, even for a non-degenerate state. The density will then not have the symmetry of the molecule, with various weird consequences.

HF is an energy-minimization variational approach, so it can “cheat” on symmetry and find lower-energy solutions. Notorious problem found often in radicals (allyl, mixed-oxidation states of bimetallic systems, many others). “Spin contamination” is a related phenomenon which we won’t discuss today. What can we do about it?

Use density-functional theory, which usually avoids it, but often has the “opposite” problem. This is precisely what is seen for BNB.

Use “multireference” methods. Often said, but this is not a panacea, and exactly the same sort of problems can occur there. A useful path to go down, but one that needs to be walked with care. Many alligators and perils to be found.

Completely avoid starting out with a zeroth-order description of the state of in the first place. Sounds peculiar, but such methods (Greens’ functions and propagator methods) are well-known in many-body physics. Coupled Cluster Theory

The wavefunction:

Schrödinger equation:

Transformed Hamiltonian:

Coupled-cluster equations: Coupled Cluster Theory THIS is the problem! The wavefunction:

Schrödinger equation:

Transformed Hamiltonian:

Coupled-cluster equations: Equation-of-motion coupled-cluster theory

I hear that term all the time. What is it, exactly?

Coupled-cluster theory suffers from symmetry breaking issues, so why would adding three words at the front make any difference?

EOM-CC is parametrized with coupled-cluster calculations, but is not coupled-cluster theory in a strict sense. It is, rather, a form of configuration interaction that use a “dressed” Hamiltonian, scales properly with the size of the system (usually) and can be improved systematically.

Similar conceptually to Green’s function methods.

Completely avoids the symmetry breaking issue. Solve coupled-cluster equations for reference state

Form

-1 electron Project onto +1 electron basis

EOMIP-CC EOMEE-CC EOMEA-CC Solve coupled-cluster BNB- equations for reference state

Form

-1 electron Project onto +1 electron basis BNB

EOMIP-CC EOMEE-CC EOMEA-CC 2 Σu state of BNB

EOMIP-CCSD ANO0 basis But what are the physics underlying all of this?

Potential energy surface distortion is the Born-Oppenheimer manifestation of “vibronic coupling”

coupling

Where is the perturber? Can we seriously expect to describe this situation without explicitly considering it?

Two-by-two matrix review The adiabatic potential energy surfaces are very nearly equal to eigenvalues of the matrix:

with

Only two parameters for the “complicated” coordinate Strongly suggests that whatever electronic basis admits to this representation might be a very useful qualitative and interpretive tool for studying the spectroscopy.

What is this basis? Simplest representation of V: The “linear vibronic coupling” model

For BNB: